Characterizing quasiconvex functions in terms of the Kronrod's tree of a function

As an application of Kronrod's construction of the tree of a function to convex analysis, a characterization of quasiconvex functions is obtained. Namely, a function is quasiconvex if and only if the associated function defined on the tree of the original function is quasiconvex. A new proof of one of the existence lemmas in Kronrod's construction of the tree of a function (the original proof of which turned out to be incorrect) is given.
Bibliography: 4 titles.